Bessel Funcs: Proving J1/uJ0 + K1/wK0 & k12J1/uJ0 + k22K1/wK0

In summary, the equations given are two linear equations with two unknowns, and the key to solving them is using the elimination method. By using this method, it can be proven that the equations are equivalent.
  • #1
Gogsey
160
0
Hi,

This is a question about modes in step index fibers, however its just the math in the following equations that I'm having trouble with, so you don't need to know the question.
basically we have the following 2 equations:

J1/uJ0 + K1/wK0 = 0

k12J1/uJ0 + k22K1/wK0 = 0

where Jn is a first order Bessel function, and Kn is a second order Bessel function.

We have to prove this but I have no idea how to do it. I've looked up various Bessel functions for all the Jn and Kn values so I could input them into the equation, but its so complicated and there's multiple expressions for them, I don't think that's what we are supposed to do.
 
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  • #2
Any help would be greatly appreciated!The key to solving this is to recognize that the two equations are linear equations with two unknowns, J1/uJ0 and K1/wK0. This means that we can solve for both unknowns using the same technique.To do so, we can use elimination, which is a method of solving linear equations by adding or subtracting them in order to eliminate one of the unknowns. For example, if we multiply the first equation by k22 and the second equation by uJ0 and then add the two equations together, the k22 term will cancel out and we will have an equation with only one unknown (K1/wK0). We can then solve for this unknown and substitute it back into either of the original equations to solve for the other unknown.Once you have done this, you should be able to prove that the two equations are equivalent.
 

Related to Bessel Funcs: Proving J1/uJ0 + K1/wK0 & k12J1/uJ0 + k22K1/wK0

1. What are Bessel functions?

Bessel functions are a type of special function in mathematics that were first introduced by the mathematician Daniel Bernoulli in 1732. They are named after the German astronomer and mathematician Friedrich Bessel who studied them extensively in the 19th century. Bessel functions have various applications in physics and engineering, particularly in problems involving wave phenomena.

2. What is the significance of proving J1/uJ0 + K1/wK0 & k12J1/uJ0 + k22K1/wK0?

This expression is known as the Riemann-Hilbert problem and is used to solve certain types of differential equations. By proving this expression, we can better understand the behavior of Bessel functions and their relationship to other mathematical functions. It also allows us to solve a wider range of problems in fields such as physics, engineering, and applied mathematics.

3. How are Bessel functions used in physics?

Bessel functions are used in physics to describe various physical phenomena, such as the propagation of electromagnetic waves, heat conduction, and quantum mechanics. They are also commonly used in solving problems involving oscillatory systems, such as vibrations and waves. Bessel functions have also been applied in signal processing and image reconstruction.

4. Can Bessel functions be expressed in closed form?

Yes, some Bessel functions can be expressed in closed form, meaning they can be written using a finite number of standard mathematical operations. However, there are many cases where Bessel functions cannot be expressed in closed form and must be approximated numerically.

5. What is the relationship between Bessel functions and other special functions?

Bessel functions are closely related to other special functions, such as hypergeometric functions and confluent hypergeometric functions. They also have connections to other mathematical concepts, such as orthogonal polynomials and the gamma function. Bessel functions are often used in combination with these other functions to solve complex mathematical problems.

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